Uniqueness for an ill-posed reaction-dispersion model.
Application to organic pollution in stream-waters

We are concerned with the inverse problem of detecting
sources
in a coupled diffusion-reaction
system. This problem arises from the
Biochemical Oxygen Demand-Dissolved
Oxygen model($^1$) governing
the interaction between organic pollutants
and the oxygen available in stream waters.
The sources we consider are point-wise and
simulate stationary or moving pollution sources.
The ultimate objective is
to
obtain their discharge location and recover their output
rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain.
It is, as a matter of fact, the most realistic configuration.
The subject to address here is the
identifiability of these sources, in other words to determine if the observations
uniquely determine the sources.
The key tool is the
study of coupled parabolic systems derived after restricting the global
model to regions at the exterior of the observations.
The absence of any prescribed
condition on the BOD density is compensated
by data recorded on the DO which provide
over-determined Cauchy boundary conditions. Now, the first
step toward the identifiability of the sources
is precisely to recover the BOD at the observation points (of DO).
This may be achieved by handling and solving
the coupled systems. Unsurprisingly, they turn out to be ill-posed.
That issue is investigated first. Then, we state
a uniqueness result owing to a suitable saddle-point variational
framework and
to Pazy's uniqueness Theorem. This uniqueness complemented by
former identifiability results proved in [2011, Inverse problems] for scalar
reaction-diffusion equations
yields the desired
identifiability for the global model.

G. Gripenberg, S.-O. London and O. Steffans, "Volterra Integral and Functional Equations,'', Encyclopedia of Mathematics and its Applications, 34 (1990).

[16]

A. Hamdi, Identification of a time-varying point source in a system of two coupled linear diffusion-advection-reaction equations: Application to surface water pollution,, Inverse Problems, 25 (2009).
doi: 10.1088/0266-5611/25/11/115009.

[17]

A. Hamdi, The recovery of a time-dependent point source in a linear transport equation: Application to surface water pollution,, Inverse Problems, 25 (2009).
doi: 10.1088/0266-5611/25/7/075006.

G. Gripenberg, S.-O. London and O. Steffans, "Volterra Integral and Functional Equations,'', Encyclopedia of Mathematics and its Applications, 34 (1990).

[16]

A. Hamdi, Identification of a time-varying point source in a system of two coupled linear diffusion-advection-reaction equations: Application to surface water pollution,, Inverse Problems, 25 (2009).
doi: 10.1088/0266-5611/25/11/115009.

[17]

A. Hamdi, The recovery of a time-dependent point source in a linear transport equation: Application to surface water pollution,, Inverse Problems, 25 (2009).
doi: 10.1088/0266-5611/25/7/075006.